In this paper an attempt has been made to estimate the parameters of Gielis superformula (modified by various functions). Simulated data have been used for this purpose. The estimation has been done by the method of simulated annealing (Download). It has been found that the simulated annealing method is quite successful in fitting the modified Gielis curves to observed data and, at the problem at hand, it performs better than the Genetic Algorithm (Download the paper). Our experiments have shown that it also performs better than the Generalized Simulated Annealing method of Tsallis and Stariolo (Download the paper : A Comparative Study on Fitting of Gielis Curves by Classical versus Generalized Simulated Annealing Methods). We have also found that the (Repulsive) Particle Swarm method of Global Optimization is more or less comparable to the Simulated Annealing method (Download the paper : Some Experiments on Fitting of Gielis Curves by Simulated Annealing and Particle Swarm Methods of Global Optimization). An attempt has also been made to estimate the parameters of the Chacon-Gielis superformula - which is a generalization of the original Gielis superformula. Chacon suggested to use Jacobian elliptic functions in place of trigonometric functions in the superformula (see Least Squares Fitting of Chacon-Gielis Curves by the Particle Swarm Method of Optimization). However, lack of empirical uniqueness of Gielis parameters has been corroborated. Due to this, it is quite unlikely to succeed at an estimation of the true parameters of Gielis superformula, more so when it is modified by an unknown function, and seek a scientific explanation behind them.
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Keywords : Gielis super-formula, supershapes, Simulated annealing, nonlinear programming, multiple sub-optimum, global, local optima, genetic algorithm, MJ Box algorithm, Nelder-Mead, fit, data, empirical, estimation, parameters, curve fitting, Simulated annealing, Particle Swarm, Elliptic function